- Chapter 1: Equations and Inequalities
- Chapter 1.1: Graphs and Graphing Utilities
- Chapter 1.2: Linear Equations and Rational Equations
- Chapter 1.3: Models and Applications
- Chapter 1.4: Complex Numbers
- Chapter 1.5: Quadratic Equations
- Chapter 1.6: Other Types of Equations
- Chapter 1.7: Linear Inequalities and Absolute Value Inequalities
- Chapter 2: Functions and Graphs
- Chapter 2.1: Basics of Functions and Their Graphs
- Chapter 2.2: More on Functions and Their Graphs
- Chapter 2.3: Linear Functions and Slope
- Chapter 2.4: More on Slope
- Chapter 2.5: Transformations of Functions
- Chapter 2.6: Combinations of Functions; Composite Functions
- Chapter 2.7: Inverse Functions
- Chapter 2.8: Distance and Midpoint Formulas; Circles
- Chapter 3: Polynomial and Rational Functions
- Chapter 3.1: Quadratic Functions
- Chapter 3.2: Polynomial Functions and Their Graphs
- Chapter 3.3: Dividing Polynomials; Remainder and Factor Theorems
- Chapter 3.4: Zeros of Polynomial Functions
- Chapter 3.5: Rational Functions and Their Graphs
- Chapter 3.6: Polynomial and Rational Inequalities
- Chapter 3.7: Modeling Using Variation
- Chapter 4: Exponential and Logarithmic Functions
- Chapter 4.1: Exponential Functions
- Chapter 4.2: Logarithmic Functions
- Chapter 4.3: Properties of Logarithms
- Chapter 4.4: Exponential and Logarithmic Equations
- Chapter 4.5: Exponential Growth and Decay; Modeling Data
- Chapter 5: Systems of Equations and Inequalities
- Chapter 5.1: Systems of Linear Equations in Two Variables
- Chapter 5.2: Systems of Linear Equations in Three Variables
- Chapter 5.3: Partial Fractions
- Chapter 5.4: Systems of Nonlinear Equations in Two Variables
- Chapter 5.5: Systems of Inequalities
- Chapter 5.6: Linear Programming
- Chapter 6: Matrices and Determinants
- Chapter 6.1: Matrix Solutions to Linear Systems
- Chapter 6.2: Inconsistent and Dependent Systems and Their Applications
- Chapter 6.3: Matrix Operations and Their Applications
- Chapter 6.4: Multiplicative Inverses of Matrices and Matrix Equations
- Chapter 6.5: Determinants and Cramer's Rule
- Chapter 7: Conic Sections
- Chapter 7.1: The Ellipse
- Chapter 7.2: The Hyperbola
- Chapter 7.3: The Parabola
- Chapter 8: Sequences, Induction, and Probability
- Chapter 8.1: Sequences and Summation Notation
- Chapter 8.2: Arithmetic Sequences
- Chapter 8.3: Geometric Sequences and Series
- Chapter 8.4: Mathematical Induction
- Chapter 8.5: The Binomial Theorem
- Chapter 8.6: Counting Principles, Permutations, and Combinations
- Chapter 8.7: Probability
- Chapter P: Prerequisites: Fundamental Concepts of Algebra
- Chapter P.1: Algebraic Expressions, Mathematical Models, and Real Numbers
- Chapter P.2: Exponents and Scientific Notation
- Chapter P.3: Radicals and Rational Exponents
- Chapter P.4: Polynomials
- Chapter P.5: Factoring Polynomials
- Chapter P.6: Rational Expressions
College Algebra 6th Edition - Solutions by Chapter
Full solutions for College Algebra | 6th Edition
Tv = Av + Vo = linear transformation plus shift.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Invert A by row operations on [A I] to reach [I A-I].
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Outer product uv T
= column times row = rank one matrix.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.